Defect bound by equivalent matrices conjecture

Let $H$ be a complex Hadamard matrix, let $d(H)$ denote its defect, and let $|1\in\widetilde{H}|$ be the number of entries equal to $1$ in a matrix $\widetilde{H}$. Two complex Hadamard matrices are equivalent when one is obtained from the other by the equivalence operations of Definition 1.3.

Defect bound conjecture. We have

$$d(H)\leq\max|1\in\widetilde{H}|,$$

where the maximum ranges over all matrices $\widetilde{H}$ equivalent to $H$.

The claim seeks to bound the deformation-theoretic defect using the largest possible number of unit entries among equivalent representatives. The source presents it as part of a broader attempt to recover the defect from the statistics of these unit-entry counts; its resolution is not given.